Rome has many large and vibrant markets held in the city centre that are well worth a visit. They usually open from 7am-1pm, Monday to Saturday although Via Portuense and Via Ippolito Nievo open on Sundays too.
Rome's oldest market is held on Campo Dei Fiori (Flora's Courtyard, not 'flower market'). Here you can find everything from the best fish in the city, fresh vegetables and attractively-piled fruit, spice, to kitchen utensils, tablecloths and toys.
Via Portuense and Via Ippolito Nievo is the main flea market in Rome and is held at the back of Trastevere quarter. Here you'll be able to find a bit of everything, though you are unlikely to find a real bargain. They also have terracotta copies of antiques and even some traditional oil lamps. Of course you should haggle, if only for fun.
Piazza dei Partigiani is a small flea market and is held in the basement of a garage and sells furniture and other objects from 1940s and 1950s. You might even come across middle class housewives from wealthy districts selling family items. The prices are usually incredibly low, a style Italians refer to as modernariato.
Via Crispi is a fairly new flea market that is held in a huge 4-floor garage between Piazza del Popolo and Via Veneto. There are many surprising things to be found here, including a certain Mario Ambrosini, a screenwriter who worked with Fellini, who sells the miniature gifts you find in Kinder chocolate eggs to collectors and members of the Kindermania club. Prices do vary and are sometimes a little on the steep side so polish your haggling skills.